# Every p-subgroup is well-placed in some Sylow subgroup

## Statement

### Statement that involves choosing a Sylow subgroup

Suppose $G$ is a finite group, $p$ is a prime number, and $W$ is a Conjugacy functor (?) on the $p$-subgroups of $G$. Suppose $H$ is a $p$-subgroup of $G$. Then, there exists a $p$-Sylow subgroup $P$ of $G$ such that $H$ is a Well-placed subgroup (?) in $P$ relative to the conjugacy functor $W$.

### Statement that involves conjugating the subgroup

Suppose $G$ is a finite group, $p$ is a prime number, and $W$ is a Conjugacy functor (?) on the $p$-subgroups of $G$. Suppose $H$ is a $p$-subgroup of $G$ contained in a $p$-Sylow subgroup $Q$. Then, there exists a subgroup $K$ of $G$, that is well-placed in $Q$, and is conjugate to $H$ in $G$.

## Facts used

1. Conjugacy functor sends every subgroup to a normalizer-relatively normal subgroup: For any $p$-subgroup $H$, $H \le N_G(W(H))$.
2. Sylow subgroups exist
3. Sylow implies order-dominating

## Proof

Given: A finite group $G$, a prime $p$, a conjugacy functor $W$ on the $p$-subgroups. A $p$-subgroup $H$ of $G$.

To prove: There exists a $p$-Sylow subgroup $P$ of $G$ such that $H$ is well-placed in $G$ relative to $W$.

Proof: Define the following ascending chain of subgroups $H_i$. $H_0 = H$, and $H_{i+1}$ is a $p$-Sylow subgroup of $N_G(W(H_i))$. By fact (1), this chain of subgroups is an ascending chain of $p$-subgroups. Let $P$ be a $p$-Sylow subgroup of $G$ containing the union of the $H_i$s (such a $P$ exists by facts (2) and (3)). We now prove that $H$ is well-placed in $P$. This follows easily from the definition.